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\journal{Nuclear Physics B}



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\title{Ferromagnetic two-component Bose gas in one dimension}

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\begin{abstract}
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The main purpose of this note is to evaluate the two-component 
Bose gas in one dimension
 via thermodynamic Bethe ansatz (TBA) equations. 
The scaling functions and the phase diagram were provided. 
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\section{The Thermodynamic Bethe Ansatz Equations for Spinor Boson
Gases}

Consider $N$ bosons of mass $m$ constrained by periodic boundary 
conditions to a line of length $L$ with internal spin degrees, 
the Hamiltonian for this system is 
\begin{align}
\mathcal{H} = 
-\frac{\hbar^2}{2m}\sum_{i=1}^{N} \frac{\partial^2}{\partial x_i^2} 
+ c \sum_{i<j} \delta (x_i - x_j),
\label{eqHamiltonian}
\end{align}
here $c = 2/a_\text{1D}$ is the scattering strength decides by the 
scattering length $a_\text{1D}$ in one dimension \cite{Olshanii1998},
and usually we set $\frac{\hbar^2}{2m} = 1$ for convenience. 
For $M$ spin-down bosons, the Bethe ansatz equations (BAE) are of 
the form
\begin{align}
& \mathrm{e}^{\mathrm{i} k_j L} 
= -\prod_{l = 1}^{N} \frac{k_j - k_l + \mathrm{i}c}{k_j - k_l - \mathrm{i}c}
\prod_{\alpha = 1}^{M} 
\frac{k_j - \lambda_\alpha - \frac{1}{2}\mathrm{i}c}
{k_j - \lambda_\alpha + \frac{1}{2}\mathrm{i}c}
\notag \\
& \prod_{l = 1}^{N} \frac{\lambda_\alpha - k_l - \frac{1}{2}\mathrm{i}c}
{\lambda_\alpha - k_l + \frac{1}{2}\mathrm{i}c}
= - \prod_{\beta = 1}^{M}
\frac{\lambda_\alpha - \lambda_\beta - \mathrm{i}c}
{\lambda_\alpha - \lambda_\beta + \mathrm{i}c},
\end{align}
for $j = 1,\cdots,N$ and $\alpha = 1,\cdots,M$.
The energy spectrum is given by $E = \sum_{i=1}^{N} k_j^2$.
From the Bethe ansatz equations, the thermodynamic Bethe ansatz
equations (TBA) can be established \cite{Guan2007,Takahashi2005}
\begin{align}
\epsilon(k) &= k^2 - \mu - \frac{1}{2}H + a_2 \ast F[\epsilon]
-T\sum_{n=1}^{\infty}a_n \ast \log(1+\eta_n^{-1})
\notag \\
\log \eta_n (k) &= \frac{nH}{T} - T a_n \ast F[\epsilon]
+ \sum_{m=1}^{\infty} T_{nm} \ast \log(1+\eta_m^{-1})
\label{eqTBA}
\end{align}
where $\mu$ is the chemical potential, $H$ is the magnetic field, 
$T$ is the temperature. Notice that some notations were introduced to make
the TBA equations simple, thus in (\ref{eqTBA}) 
``$\ast$" denote the convolution product 
\begin{align}
(f \ast g)(k) \triangleq \int_{-\infty}^{\infty} \mathrm{d} p f(k-p)g(p)
\end{align}
and $F$ denotes the function 
\begin{align}
F[x] \triangleq -T\log(1+\mathrm{e}^{-\frac{x}{T}}).
\end{align}
The functions $a_n$ and $T_{nm}$ are introduced in \cite{Guan2007}
\begin{align}
a_n(x) = \frac{1}{2\pi}\frac{nc}{(nc/2)^2 + x^2}
\end{align}
for $c$ is the scattering constant in (\ref{eqHamiltonian}) and
\begin{align}
T_{nm} &= \left\{
\begin{array}{ll}
a_{\mid m-n \mid} + 2a_{\mid m-n \mid + 2} + \cdots + 2a_{m+n-2} + a_{m+n}, 
&  \text{if} \;\; m \neq n, \\
2a_2 + 2a_4 + \cdots + 2a_{2n-2} + a_{2n}, &  \text{if} \;\; m = n.
\end{array}\right.
\end{align}
It was showed in \cite{Guan2007,Takahashi2005} that the $\eta_n(k)$
string satisfies the asymptotic behaviour 
\begin{align}
\eta_n^{\infty} \triangleq \lim_{k \to \infty} \eta_n (k) = \left[
\frac{\sinh \frac{(n+1)H}{2T}}{\sinh \frac{H}{2T}}  \right]^2 - 1,
\end{align} 
thus if we make the substitution
\begin{align}
u_n(k) = \log \eta_n(k) - \log \eta_n^{\infty},
\end{align}
the TBA equations (\ref{eqTBA}) arrive at the new form
\begin{align}
\epsilon(k) &= k^2 - \mu - T\log(2\cosh\frac{H}{2T})
+ a_2 \ast F[\epsilon] 
- T\sum_{n=1}^{\infty} a_n \ast 
\log\frac{\eta_n^{\infty}+\mathrm{e}^{-u_n}}{\eta_n^{\infty} + 1},
\notag \\
u_n(k) &= -\frac{1}{T}a_n \ast F[\epsilon] 
+ \sum_{m=1}^{\infty} T_{nm} \ast
\log\frac{\eta_m^{\infty}+\mathrm{e}^{-u_m}}{\eta_m^{\infty} + 1}.
\label{eqTBA2}
\end{align}

\subsection{The pressure}

The pressure $p(T,H,\mu,c)$ can be calculated via
the solution $\epsilon$ in (\ref{eqTBA}) or (\ref{eqTBA2})
\begin{align}
p = -\int_{-\infty}^{\infty} \frac{\mathrm{d}k}{2\pi}
 \, F[\epsilon](k),
\label{eqPressure}
\end{align}
and the thermal potential for the grand canonical ensemble is
$\Omega = -pL$.
The TBA functions (\ref{eqTBA2}) together with
 the pressure expression (\ref{eqPressure})
are complete to determine the full statistic properties for 
the model (\ref{eqHamiltonian}) we are interested in.  

\subsection{The polarization}
Usually the detection of the phase transitions for 
the spin system relies on the polarization of the spins. 
The polarization is determined by the derivation of the 
grand thermal potential by the magnetic field
\begin{align}
m_z = \frac{M_z}{N} = \frac{\partial p}{\partial H}
\end{align}
and according to (\ref{eqPressure}),
\begin{align}
\frac{\partial p}{\partial H} = -\int_{-\infty}^{\infty} 
\frac{\mathrm{d}k}{2\pi}
\frac{\partial \epsilon}{\partial H}
 \, G[\epsilon](k),
\label{eqPressure_dh}
\end{align}
where the function $G[\epsilon]$ is defined by
\begin{align}
G[\epsilon] \triangleq \frac{1}{1+\mathrm{e}^{\frac{x}{T}}}.
\end{align}
The derivation of the TBA equations (\ref{eqTBA2}) reads
\begin{align}
\frac{\partial \epsilon}{\partial H} 
&= -\frac{1}{2}\tanh \frac{H}{2T}
+ a_2 \ast G[\epsilon] \frac{\partial \epsilon}{\partial H} 
- T\sum_{n=1}^{\infty} a_n \ast 
\left( P_n - Q_n \frac{\partial u_n}{\partial H} \right)
\notag \\
\frac{\partial u_n}{\partial H} &= 
-\frac{1}{T}a_n \ast G[\epsilon]\frac{\partial \epsilon}{\partial H} 
+ \sum_{m=1}^{\infty} T_{nm} \ast
\left( P_m - Q_m \frac{\partial u_m}{\partial H} \right),
\end{align}
where the functions $P_n(k)$ and $Q_n(k)$ are defined by
\begin{align}
P_n(k) & \triangleq 
\frac{1-\mathrm{e}^{-u_n(k)}}{T} \cdot
\frac{2\left( (n+1)\coth[\frac{H}{2T}(n+1)]-\coth\frac{H}{2T}\right)\sinh^2\frac{H}{2T}}
{\cosh[\frac{H}{T}(n+1)]- \mathrm{e}^{-u_n(k)}-(1-\mathrm{e}^{-u_n(k)})\cosh\frac{H}{T} }
\notag \\
Q_n(k) & \triangleq
\frac{1}{\eta_n^\infty \mathrm{e}^{u_n(k)} + 1}.
\end{align}
Notice that the $P_n(k) \to 0$ when $\frac{H}{T} \to 0$ or $\frac{H}{T} \to \infty$.




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